Developmental Math – An Open Curriculum Instructor Guide
Unit 12 – Table of Contents
Unit 12: Introduction to Factoring
Learning Objectives 12.2
Instructor Notes 12.3 The Mathematics of Factoring
Teaching Tips: Challenges and Approaches Additional Resources
Instructor Overview 12.10 Tutor Simulation: Playing the Elimination Game
Instructor Overview 12.11 Puzzle: Match Factors
Instructor Overview 12.13 Project: Making Connections
Common Core Standards 12.43
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12.1 Developmental Math – An Open Curriculum Instructor Guide
Unit 12 – Learning Objectives
Unit 12: Factoring
Lesson 1: Introduction to Factoring
Topic 1: Greatest Common Factor Learning Objectives Find the greatest common factor (GCF) of monomials. Factor polynomials by factoring out the greatest common factor (GCF). Factor expressions with four terms by grouping.
Lesson 2: Factoring Polynomials
Topic 1: Factoring Trinomials Learning Objectives Factor trinomials with a leading coefficient of 1. Factor trinomials with a common factor. Factor trinomials with a leading coefficient other than 1. Topic 2: Factoring: Special Cases Learning Objectives Factor trinomials that are perfect squares. Factor binomials in the form of the difference of squares. Topic 3: Special Cases: Cubes Learning Objectives Factor the sum of cubes. Factor the difference of cubes.
Lesson 3: Solving Quadratic Equations
Topic 1: Solve Quadratic Equations by Factoring Learning Objectives Solve equations in factored form by using the Principle of Zero Products. Solve quadratic equations by factoring and then using the Principle of Zero Products. Solve application problems involving quadratic equations.
12.2 Developmental Math – An Open Curriculum Instructor Guide
Unit 12 – Instructor Notes
Unit 12: Factoring
Instructor Notes
The Mathematics of Factoring
This unit builds upon students’ knowledge of polynomials learned in the previous unit. They will learn how to use the distributive property and greatest common factors to find the factored form of binomials and how to factor trinomials by grouping. Students will also learn how to recognize and quickly factor special products (perfect square trinomials, difference of squares, and the sum and difference of two squares). Finally, they’ll get experience combining these techniques and using them to solve quadratic equations.
Teaching Tips: Challenges and Approaches
This unit on factoring is probably one of the most difficult—students will spend a lot of time carrying out multi-step, complex procedures for what will often seem to be obscure purposes. At this stage in algebra, factoring polynomials may feel like busy work rather than a means to a useful end. It doesn’t help that students may remember having trouble with factoring from when they studied algebra in high school.
Encourage students to think of factoring as the reverse of multiplying polynomials that was learned previously. Then, a problem multiplying polynomials was given and students were asked to calculate the answer. In this unit, the answer is given and the students need to come up with the question. Sound familiar? In a way, factoring is like playing the popular game show Jeopardy.
Greatest Common Factor Finding the greatest common factor of whole numbers should be reviewed before finding the GCF of polynomials. Then it is a logical step to demonstrate how to factor expressions by using the distributive property in reverse to pull out the greatest common monomial from each term in a polynomial:
12.3 Developmental Math – An Open Curriculum Instructor Guide
[From Lesson 1, Topic 1, Topic Text]
Remind your students to pay particular attention to signs as it is easy to make a mistake with them, and also to check their final answers by multiplying.
Grouping After your students are comfortable pulling the GCF out of a polynomial, it is time to teach them a new method of factoring–factoring by grouping. Begin by introducing the technique on 4-term polynomials. It's fairly easy for students to understand how to break these polynomials into groups of two and then factor each pair.
Trinomials are trickier. Indeed, many textbooks do not use grouping for factoring trinomials, and instead use essentially a guess and check method. While factoring by grouping may initially be a more complex procedure, it has many significant advantages in the long term and is used in this course. The hardest part is figuring out how to rewrite the middle term of a trinomial as an equivalent binomial. Students will need to see this demonstrated repeatedly, as well as get a lot of practice working on their own. Even after they grasp the basic idea, they'll often have trouble deciding which signs to use. It will be helpful to supply them with a set of tips like the one below:
12.4 Developmental Math – An Open Curriculum Instructor Guide
[From Lesson 2, Topic 1, Topic Text]
Factoring by grouping has the great advantage of working for all trinomials. It also provides a method to determine when a polynomial cannot be factored. (This is not obvious when students are using the guess and check method.)
Sometimes students don’t remember to look for the greatest common factor of all the terms of a polynomial before trying to factor by grouping. This isn’t wrong, but the larger numbers can make the work more difficult. Plus the student has to remember to look for a greatest common factor at the end anyway. In order to illustrate this, have students factor 9x2 + 15x − 36 without pulling out the greatest common factor of 3 -- they will notice that the numbers are cumbersome. After this, have them try again, this time factoring out the 3 as the first step. They will see the benefits.
Once the grouping method is mastered, let your students use it to factor perfect square trinomials. Hopefully they'll soon see a pattern, though you will probably have to nudge them along. Eventually, they should learn to recognize if a trinomial is a perfect square, and be able to factor it without grouping.
After the rule for factoring a perfect square trinomial has been developed, set them to finding one for factoring the difference of two squares. This rule is usually very easy for students to figure out. Then have them try to factor the sum of two squares, such as x2 + 4. Make sure they understand that this cannot be done.
12.5 Developmental Math – An Open Curriculum Instructor Guide Intermediate algebra students will also need to know how to factor the sum and difference of two cubes. They are sure to have trouble remembering the formulas. Try pointing out that the formulas are really the same except for signs:
A binomial in the form a3 + b3 can be factored as (a + b)(a2 – ab + b2) A binomial in the form a3 – b3 can be factored as (a – b)(a2 + ab + b2)
The sign in between the two cubes is the same sign as in the first factor in the formulas. The next sign is the opposite of the first sign and the last sign is always positive. Now “all” they have to remember are the variable parts of the formulas. Easy!
Factoring Quadratic Equations The last topic in this unit is solving quadratic equations by factoring and applying the zero products rule. Begin by solving an example where the polynomial is already factored and set equal to zero, such as the following:
[From Lesson 3, Topic 1, Worked Example 1]
Now give your students a problem like "Solve x2 + x – 12 = 0 for x." Ask them how they would attempt to solve for x. Someone will suggest factoring the left hand side by grouping and they will be on their way.
Then pose the problem x2 + x – 12 = 18. Make sure your students know that in order for the principle of zero products to work, the trinomial must be set equal to 0. Sometimes students are
12.6 Developmental Math – An Open Curriculum Instructor Guide so focused on new techniques, they forget basic principles for rewriting an equation and they may need to be prodded to add (or subtract) something to (or from) both sides so that one side equals zero.
Be careful -- once students get into the hang of applying the zero products rule to solve equations, they may start trying it on expressions as well. For instance, if a problem says to factor x2 + x – 12, some will do so and then go ahead and calculate that x = -4 or 3. Remind your students to only do what a problem asks – factor when it says to factor and solve when it says to solve.
The Sense Test Application problems have an extra requirement that solving given equations do not -- answers have to make sense based on their context. Consider the following scenario:
[From Lesson 3, Topic 1, Topic Text]
12.7 Developmental Math – An Open Curriculum Instructor Guide 1 Mathematically, it is true that t can be either 4 or . But logically, only one of these answers 2 works -- since t represents the number of seconds after the rocket has taken off, it can’t be a negative number. The rocket can't hit the ground before it was launched. Teach students that when they do application problems like this, they need to check not only the math but also the sense of their results.
Keep in Mind
Factoring trinomials and solving quadratic equations are difficult topics. As soon as you say “factoring,” some students will recall hours of erasing before correct answers were found through trial and error. Reassure students that while the factoring by grouping method takes longer to use when working simple problems, it really will make solving complex problems quicker. Stress to your students that once something is factored, they should check their work by multiplying. This will help them catch any errors that were made.
Most of the material in this unit has been geared to both beginning and intermediate students. More difficult examples and problems are included for the intermediate algebra student, but these could be used to challenge the beginning algebra student. However, two topics, factoring the sum and difference of two cubes, are intended only for intermediate algebra students.
Additional Resources In all mathematics, the best way to really learn new skills and ideas is repetition. Problem solving is woven into every aspect of this course—each topic includes warm-up, practice, and review problems for students to solve on their own. The presentations, worked examples, and topic texts demonstrate how to tackle even more problems. But practice makes perfect, and some students will benefit from additional work.
Practice finding the common factor of polynomials at http://www.mathsnet.net/algebra/a41.html (get additional problems by clicking on “more on this topic”).
Factoring practice using the AC Method can be found at http://www.ltcconline.net/greenl/java/BasicAlgebra/AC/AC.html.
Solve quadratic equations using the principle of zero products at http://www.mathsnet.net/algebra/e34.html (get additional problems by clicking on “more on this topic”).
Practice all types of factoring problems at http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=a lgebra&s2=factor&s3=basic.
Review factoring and solving quadratic equations at http://www.quia.com/rr/36611.html.
12.8 Developmental Math – An Open Curriculum Instructor Guide
Summary
After completing this unit, students will be more comfortable with factoring any polynomial that is given to them. They'll be able to pull out the GCF and factor by grouping, and recognize special cases such as perfect square trinomials, difference of two squares, and the sum and difference of two cubes. They'll have had experience combining these techniques to solve quadratic equations, and will have gained an appreciation that factoring can be used to solve real-life problems.
12.9 Developmental Math – An Open Curriculum Instructor Guide
Unit 12 – Tutor Simulation
Unit 12: Factoring
Instructor Overview Tutor Simulation: Playing the Elimination Game
Purpose
This simulation allows students to demonstrate their ability to factor polynomials. Students will be asked to apply what they have learned to solve a problem involving:
Factoring out the greatest common factor Factoring by grouping Factoring the sum or difference of perfect squares Factoring the sum or difference of cubes
Problem Students are presented with the following problem:
Many mathematicians have tricks they use to analyze an expression in order to determine if they can factor it quickly. In this simulation, we are going to focus on techniques for quickly factoring polynomials with two terms (binomials) or with four terms.
Recommendations Tutor simulations are designed to give students a chance to assess their understanding of unit material in a personal, risk-free situation. Before directing students to the simulation,
Make sure they have completed all other unit material. Explain the mechanics of tutor simulations. o Students will be given a problem and then guided through its solution by a video tutor; o After each answer is chosen, students should wait for tutor feedback before continuing; o After the simulation is completed, students will be given an assessment of their efforts. If areas of concern are found, the students should review unit materials or seek help from their instructor. Emphasize that this is an exploration, not an exam.
12.10 Developmental Math – An Open Curriculum Instructor Guide
Unit 12 – Puzzle
Unit 12: Factoring
Instructor Overview Puzzle: Match Factors
Objectives
Match Factors is a puzzle that tests a player's ability to factor by grouping. It reinforces the technique of factoring a trinomial in the form ax2 + bx + c by finding two integers, r and s, whose sum is b and whose product is ac. Puzzle play, especially when done by eye rather than with pencil and paper, will help students learn to quickly identify the components of factors.
Figure 1. Match Factors players choose the factors of a central polynomial from a rotating ring of possibilities.
Description
Each Match Factors game consists of a sequence of 4 polynomials surrounded by 8 possible factors. As each polynomial is displayed, players are asked to pick the matching pair of factors. If they choose correctly, the next polynomial appears. If not, they must try again before play advances.
12.11 Developmental Math – An Open Curriculum Instructor Guide There are three levels of play, each containing 10 games. In Level 1, polynomials have the form x2 + bx + c. Level 2 polynomials have the form ax2 + bx + c. Players in Level 3 must factor ax2 + bxy + cy2 polynomials.
Match Factors is suitable for individual or group play. It could also be used in a classroom setting, with the whole group taking turns calling out the two factors of each expression.
12.12 Developmental Math – An Open Curriculum Instructor Guide
Unit 12 – Project
Unit 12: Factoring
Instructor Overview Project: Making Connections
Student Instructions
Introduction
The main business of science is to uncover patterns. Often we represent those patterns as algebraic expressions, graphs, or tables of numbers (data). Being able to make connections among those various representations is one of the most vital skills to possess.
Task
In this project you attempt to make precise connections among these three ways of representing patterns.
Instructions (See the online course materials for full size graphs and charts)
Work with at least one other person to complete the following exercises. Solve each problem in order and save your work along the way. You will create a presentation on one of the four parts to be given to your class.
First Problem – Connecting Algebraic Expressions and Graphs: Factor each of the following expressions completely, and then compare the factored form with the “picture” of the expression that is shown in the graph on the right. Describe any connections that you see, and then repeat for the next expression. In the end, formulate a conjecture as to how an algebraic expression in factored form is related to its corresponding graph. Keep in mind that we are not assuming that you have any knowledge whatsoever about graphs. That is what makes this “detective work” so fun!
12.13 Developmental Math – An Open Curriculum Instructor Guide
Algebraic Possible Graph Expression Relationship
2a3
Factored Form
3b 12
Factored Form
6x2 54
Factored Form
12.14 Developmental Math – An Open Curriculum Instructor Guide Algebraic Possible Graph Expression Relationship
x6x92
Factored Form
y2 10y 25
Factored Form
2c2 4c 6
Factored Form
12.15 Developmental Math – An Open Curriculum Instructor Guide Algebraic Possible Graph Expression Relationship
3a243
Factored Form
3a3 24
Factored Form
2x2 4x 30
Factored Form
12.16 Developmental Math – An Open Curriculum Instructor Guide Algebraic Possible Graph Expression Relationship
5d15d43
Factored Form
2x5 4x 43 30x
Factored Form
12.17 Developmental Math – An Open Curriculum Instructor Guide
The following problems are a special challenge. You may have to adjust the relationship you expressed above to accommodate these new examples.
2x5x2 3
Factored Form
6t2 t 2
Factored Form
Second Problem – Connecting Algebraic Expressions and Tables: For each of the same algebraic expressions that you examined above, compare the factored form with the table of values associated with the expression. (For example, if the expression is 2 2 2x 5x 3 , then the value associated with it when x=1 will be 2(1) 5(1) 3 4.) In the end, formulate a conjecture that describes how an algebraic expression in factored form is related to its corresponding data table.
12.18 Developmental Math – An Open Curriculum Instructor Guide
Algebraic Table Possible Relationship Expression
a 2a3 -8 -1024 2a3 -7 -686 -6 -432 -5 -250 -4 -128 -3 -54 -2 -16 Factored Form -1 -2 0 0 1 2 2 16 3 54 4 128 5 250 6 432 7 686 8 1024
b 3b 12
-8 -36 3b 12 -7 -33
-6 -30
12.19 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 -27
-4 -24
-3 -21
-2 -18
-1 -15
0 -12
1 -9
2 -6
3 -3
4 0
5 3
6 6
7 9
8 12
x 6x2 54
-8 330 6x2 54 -7 240
-6 162
12.20 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 96
-4 42
-3 0
-2 -30
-1 -48
0 -54
1 -48
2 -30
3 0
4 42
5 96
6 162
7 240
8 330
x x2 6x 9
-8 25 x2 6x 9 -7 16
-6 9
12.21 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 4
-4 1
-3 0
-2 1
-1 4
0 9
1 16
2 25
3 36
4 49
5 64
6 81
7 100
8 121
2 y y 10y 25
y2 10y 25 -8 169 -7 144
-6 121
12.22 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 100
-4 81
-3 64
-2 49
-1 36
0 25
1 16
2 9
3 4
4 1
5 0
6 1
7 4
8 9
c 2c2 4c 6
-8 90 2c2 4c 6 -7 64
-6 42
12.23 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 24
-4 10
-3 0
-2 -6
-1 -8
0 -6
1 0
2 10
3 24
4 42
5 64
6 90
7 120
8 154
a 3a3 24
-8 -1560 3a3 24 -7 -1053
-6 -672
12.24 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 -399
-4 -216
-3 -105
-2 -48
-1 -27
0 -24
1 -21
2 0
3 57
4 168
5 351
6 624
7 1005
8 1512
a 3a3 24
-8 -1512 3a3 24 -7 -1005
-6 -624
12.25 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 -351
-4 -168
-3 -57
-2 0
-1 21
0 24
1 27
2 48
3 105
4 216
5 399
6 672
7 1053
8 1560
x 2x2 4x 30
-8 130 2x2 4x 30 -7 96
-6 66
12.26 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 40
-4 18
-3 0
-2 -14
-1 -24
0 -30
1 -32
2 -30
3 -24
4 -14
5 0
6 18
7 40
8 66
d 5d43 15d
-8 28160 5d43 15d -7 17150
-6 9720
12.27 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 5000
-4 2240
-3 810
-2 200
-1 20
0 0
1 -10
2 -40
3 0
4 320
5 1250
6 3240
7 6860
8 12800
x 2x5 4x 4 30x 3 2x5 4x 4 30x 3 -8 -66560
-7 -32928
12.28 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -6 -14256
-5 -5000
-4 -1152
-3 0
-2 112
-1 24
0 0
1 -32
2 -240
3 -648
4 -896
5 0
6 3888
7 13720
8 33792
x 2x2 5x 3
-8 85 2x2 5x 3 -7 60
-6 39
12.29 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 22
-4 9
-3 0
-2 -5
-1 -6
0 -3
1 4
2 15
3 30
4 49
5 72
6 99
7 130
8 165
t 6t2 t 2
-8 390 6t2 t 2 -7 299
-6 220
12.30 Developmental Math – An Open Curriculum Instructor Guide Algebraic Table Possible Relationship Expression
Factored Form -5 153
-4 98
-3 55
-2 24
-1 5
0 -2
1 3
2 20
3 49
4 90
5 143
6 208
7 285
8 374
Third Problem – Applying Your Findings: For each expression, factor it completely and write the factored form beneath the expression. Then match it to its corresponding table or graph by writing the letter corresponding to the expression on its matching table or graph.
12.31 Developmental Math – An Open Curriculum Instructor Guide
54 53 a) 3c12 b) 2a9a c) 3d12d
3 3 2 d) y27 e) 2z16 f) 2x +5x-12
7 2 2 g) 3a h) t6t 7 i) 9y6y 1
12.32 Developmental Math – An Open Curriculum Instructor Guide
Variable Expression
-8 -92160
-7 -46305
-6 -20736
-5 -7875
-4 -2304
-3 -405
-2 0
-1 9
0 0
1 -9
2 0
3 405
4 2304
5 7875
6 20736
7 46305
8 92160
12.33 Developmental Math – An Open Curriculum Instructor Guide
Variable Expression Variable Expression
-8 -6291456 -8 -1040
-7 -2470629 -7 -702
-6 -839808 -6 -448
-5 -234375 -5 -266
-4 -49152 -4 -144
-3 -6561 -3 -70
-2 -384 -2 -32
-1 -3 -1 -18
0 0 0 -16
1 3 1 -14
2 384 2 0
3 6561 3 38
4 49152 4 112
5 234375 5 234
6 839808 6 416
7 2470629 7 670
8 6291456 8 1008
12.34 Developmental Math – An Open Curriculum Instructor Guide
Variable Expression
-8 -36
-7 -33
-6 -30
-5 -27
-4 -24
-3 -21
-2 -18
-1 -15
0 -12
1 -9
2 -6
3 -3
4 0
5 3
6 6
7 9
8 12
Fourth Problem – Predicting the Unknown: One of the primary reasons to make connections is to be able to explain or predict previously unobserved behavior. Below we provide you with some tables and some graphs. Based on these alone, determine whether the expression associated with them can be factored. Explain the reasoning behind your decision. [Hint: You should make use of your observations from the problems above to determine what it means for an expression to not be factorable.]
12.35 Developmental Math – An Open Curriculum Instructor Guide
Variable Expression
-8 -550
-7 -376
-6 -244
-5 -148
-4 -82
-3 -40
-2 -16
-1 -4
0 2
1 8
2 20
3 44
4 86
5 152
6 248
7 380
8 554
12.36 Developmental Math – An Open Curriculum Instructor Guide
Variable Expression
-8 344
-7 193
-6 98
-5 44
-4 18
-3 8
-2 7
-1 8
0 9
1 6
2 2
3 1
4 8
5 32
6 84
7 176
8 325
Conclusions
With those from another group, compare your answers and your way of talking about the connections between the factored form of the expressions and the graphs and tables. Work to make sure that your explanation is clear and concise.
Prepare a presentation which:
1. Explains the connection between the factored expression and the corresponding graphs and tables.
12.37 Developmental Math – An Open Curriculum Instructor Guide 2. Describes briefly how you determined this connection (you may want to discuss some of your original ideas and how you needed to refine them as you looked at more examples). 3. Gives a test for determining whether a given expression can be factored if you are given a graph or table associated with the expression.
Finally, present your solution to your instructor.
Instructor Notes We would stress that nothing in this project assumes that students have any familiarity whatsoever with graphing. In fact, it is precisely because they do not have this familiarity that we can explore this topic. The project is more about developing students’ abilities to notice connections between the various functional representations before they even understand how these representations work. So, this project can serve both as a culminating project for Unit 12 on factoring as well as very initial preparation for a unit on graphing.
Assignment Procedures Problem 1 The relationship that they should be identifying is that the values of the variable that make each factor zero will correspond to the places where the graph crosses the horizontal axis. For the first eleven graphs, the student may very well identify the “the negative of the number in the factor” as the place where the graph crosses the axis. For example, for 2c2 4c 6 = 2 (c+3) (c-1) , the graph crosses at c=-3 and c=1. However, when they encounter the last two, for example 2x2 5x 3 = (2x-1) (x+3) , the graph does indeed cross 1 at x=-3, but it crosses a second time at x= and not at x=1, even though x=1 is the 2 “negative of the number that appears in the factor.” It may be a challenge for them to determine the true connection, although the fact that they have had experience solving quadratic equations by factoring should facilitate the process.
Problem 2 The connection is that the value of the variable which makes each factor equal to zero (and therefore the entire expression equal to zero as well) is the one which corresponds to a zero value for the expression in the table. Notice that, since the values of the variable rise in increments of one, the exact values of the variable that correspond to a zero value for the expression do not actually appear in the last two tables. In these examples, the students will have to notice that the value of the expression changes sign and, therefore, must have been zero somewhere in between.
Problem 3 The following table shows the correct matching.
12.38 Developmental Math – An Open Curriculum Instructor Guide Graph: f Graph: b
Table: c Graph: h
Graph: d Graph: i
Table: g Table: e
Table: a
Problem 4 The table below shows the solutions.
Graph: not factorable since the graph Table: factorable because somewhere does not cross the horizontal axis. between x=-1 and x=0 it is equal to zero.
Graph: not factorable since the graph Graph: factorable since the graph does not cross the horizontal axis. crosses the horizontal axis somewhere between 4 and 6.
Table: not factorable since nowhere does it appear that the expression changes sign or is zero. Note that this is only speculative since the table shows data only for values of x that are integers. However, the students at this stage need not be attentive to this nuance.
At this stage it can be helpful to tell the students in each group that for each question, you will randomly choose one person in the group to present the group’s answer. This provides motivation for the group as a whole to ensure that each member has a thorough understanding of all of the topics and gives the instructor feedback on how well each individual understands the work that was completed.
Recommendations
Have students work in teams to encourage brainstorming and cooperative learning. Assign a specific timeline for completion of the project that includes milestone dates. Provide students feedback as they complete each milestone. Ensure that each member of student groups has a specific job.
Technology Integration This project provides abundant opportunities for technology integration, and gives students the chance to research and collaborate using online technology. The students’ instructions list several websites that provide information on numbering systems, game design, and graphics.
12.39 Developmental Math – An Open Curriculum Instructor Guide The following are other examples of free Internet resources that can be used to support this project: http://www.moodle.org
An Open Source Course Management System (CMS), also known as a Learning Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular among educators around the world as a tool for creating online dynamic websites for their students. http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview
Allows you to create a secure online Wiki workspace in about 60 seconds. Encourage classroom participation with interactive Wiki pages that students can view and edit from any computer. Share class resources and completed student work. http://www.docs.google.com
Allows students to collaborate in real-time from any computer. Google Docs provides free access and storage for word processing, spreadsheets, presentations, and surveys. This is ideal for group projects. http://why.openoffice.org/
The leading open-source office software suite for word processing, spreadsheets, presentations, graphics, databases and more. It can read and write files from other common office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded and used completely free of charge for any purpose.
12.40 Developmental Math – An Open Curriculum Instructor Guide
Rubric
Score Content Presentation/Communication
• The solution shows a deep understanding of • There is a clear, effective explanation the problem including the ability to identify detailing how the problem is solved. the appropriate mathematical concepts and All of the steps are included so that the information necessary for its solution. the reader does not need to infer • The solution completely addresses all how and why decisions were made. mathematical components presented in the • Mathematical representation is task. actively used as a means of 4 • The solution puts to use the underlying communicating ideas related to the mathematical concepts upon which the task solution of the problem. is designed and applies procedures • There is precise and appropriate use accurately to correctly solve the problem of mathematical terminology and and verify the results. notation. • Mathematically relevant observations and/or • Your project is professional looking connections are made. with graphics and effective use of color. • The solution shows that the student has a • There is a clear explanation. broad understanding of the problem and the • There is appropriate use of accurate major concepts necessary for its solution. mathematical representation. • The solution addresses all of the • There is effective use of mathematical components presented in the mathematical terminology and 3 task. notation. • The student uses a strategy that includes • Your project is neat with graphics mathematical procedures and some and effective use of color. mathematical reasoning that leads to a solution of the problem. • Most parts of the project are correct with only minor mathematical errors. • The solution is not complete indicating that • Your project is hard to follow parts of the problem are not understood. because the material is presented in • The solution addresses some, but not all of a manner that jumps around between the mathematical components presented in unconnected topics. the task. • There is some use of appropriate 2 • The student uses a strategy that is partially mathematical representation. useful, and demonstrates some evidence of • There is some use of mathematical mathematical reasoning. terminology and notation appropriate • Some parts of the project may be correct, to the problem. but major errors are noted and the student • Your project contains low quality could not completely carry out mathematical graphics and colors that do not add procedures. interest to the project. • There is no solution, or the solution has no • There is no explanation of the relationship to the task. solution, the explanation cannot be 1 • No evidence of a strategy, procedure, or understood or it is unrelated to the mathematical reasoning and/or uses a problem. strategy that does not help solve the • There is no use or inappropriate use problem. of mathematical representations (e.g.
12.41 Developmental Math – An Open Curriculum Instructor Guide • The solution addresses none of the figures, diagrams, graphs, tables, mathematical components presented in the etc.). task. • There is no use, or mostly • There were so many errors in mathematical inappropriate use, of mathematical procedures that the problem could not be terminology and notation. solved. • Your project is missing graphics and uses little to no color.
12.42 Developmental Math – An Open Curriculum Instructor Guide
Unit 12 – Correlation to Common Core Standards
Learning Objectives Unit 12: Factoring
Common Core Standards
Unit 12, Lesson 1, Topic 1: Greatest Common Factor Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions STANDARD Interpret the structure of expressions. EXPECTATION A- Interpret expressions that represent a quantity in terms of its SSE.1. context. GRADE EXPECTATION A- Interpret parts of an expression, such as terms, factors, and SSE.1(a) coefficients.
Unit 12, Lesson 2, Topic 1: Factor Trinomials Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions STANDARD Interpret the structure of expressions. EXPECTATION A- Interpret expressions that represent a quantity in terms of its SSE.1. context. GRADE EXPECTATION A- Interpret parts of an expression, such as terms, factors, and SSE.1(a) coefficients.
12.43 Developmental Math – An Open Curriculum Instructor Guide Unit 12, Lesson 2, Topic 2: Factoring: Special Cases Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions STANDARD Interpret the structure of expressions. EXPECTATION A- Interpret expressions that represent a quantity in terms of its SSE.1. context. GRADE EXPECTATION A- Interpret parts of an expression, such as terms, factors, and SSE.1(a) coefficients.
Unit 12, Lesson 2, Topic 3: Special Cases: Cubes Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions STANDARD Interpret the structure of expressions. EXPECTATION A- Interpret expressions that represent a quantity in terms of its SSE.1. context. GRADE EXPECTATION A- Interpret parts of an expression, such as terms, factors, and SSE.1(a) coefficients.
Unit 12, Lesson 3, Topic 1: Solve Quadratic Equations by Factoring Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them. STRAND / DOMAIN CC.A. Algebra
12.44 Developmental Math – An Open Curriculum Instructor Guide CATEGORY / CLUSTER A-SSE. Seeing Structure in Expressions STANDARD Write expressions in equivalent forms to solve problems. EXPECTATION A- Choose and produce an equivalent form of an expression to reveal SSE.3. and explain properties of the quantity represented by the expression. GRADE EXPECTATION A- Factor a quadratic expression to reveal the zeros of the function it SSE.3(a) defines. STRAND / DOMAIN CC.A. Algebra CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities STANDARD Solve equations and inequalities in one variable. EXPECTATION A-REI.4. Solve quadratic equations in one variable. GRADE EXPECTATION A- Solve quadratic equations by inspection (e.g., for x^2 = 49), taking REI.4(b) square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a plus-minus bi for real numbers a and b. STRAND / DOMAIN CC.F. Functions CATEGORY / CLUSTER F-IF. Interpreting Functions STANDARD Analyze functions using different representations. EXPECTATION F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. GRADE EXPECTATION F- Use the process of factoring and completing the square in a IF.8(a) quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
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